Arithmetical puzzle blocks



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ARITHMETICM. LOCKS.

APILICATIOII m 192:.

Patented Oct. 17, 1922.

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F. s. GREENE. AHITHMETICAI. PUZZLE BLOCKS.

APPLICATION FILED JUNE l3. 192l- 1,432,062 Patented Oct. 17, 1922.

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ARITHMEHCAI. PUZZLE BLOCKS.

APPLICATION mcu JUHE1B. 192:.

1,432,0 Patented Oct. 11,1922.

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Z 6e E 114 A 3 31am tom Patented 0st. 17, 1922 UNETED STATES FRANK S. GREENE, 0F CLEVEIJl-ND, OHIO.

' ARITHMETICAL .r'ozzLE BLooxs.

Application filed. June 18, 1921.

1/ '0 (all whom. it may concern:

Be it known that I, FRANK S. GREENE, a citizen of the United States, residing at Cleveland in the county of Cuyahoga and State of hi0, have invented a certom new and useful Improvement in Arithmetical Puzzle Blocks, of whichthc following is a full, clear and exact description, reference being had tothe accompanying drawings.

This invention relates to arithmetical block puzzles or games involving numerical arrangements of the type known as magic squares.

It is the object of this invention to present an interesting. but rather intricate mathematical problem in such a manner that the various mathematical relationships can be visually demonstrated by a. method simple enough to be readily learned by even very young children, and to encourage the exercise of the faculties of observation and de ductiou by providing uzzles or problems of gradually increasing 'ificulty, for the solution of which intelligent observation in connection with the simpler problems is essential, the series of roblems presenting all the data neccssar flir the complete solution or explanation 0 the Jrinciples upon which the puzzle is based. [ore specifically it is the object of this invention to provide a. numerical block puzzle or game involving a simple method of placing the blocks xvlth their numerals in magic square formation, the demonstration of said method providing excellent entertainment and exciting curiosity and wonder, thus providing for young children on interesting game in which they acquire proficiency in mental arithmetic and also providing a uzzle or game which up: pools to older chi dren and adults by exciting curiosity as to the underlying principles and invites systematic study thereof b providing means problems may be worked out, intelligent observation in connection with the various problems suggested, furnishing suliicicnt data on which to base a complete explanation of the whole puzzle, thus providing a puzzle or game which will provide amusement to all and as much mental exercise as the-individual cares to indulge in, the complete problem providing a rather severe test of the individuals reasoning capacity.

in the drawing forming a part of this spccificutiou,

whereby a series of distinct Serial No. 478,504.

Figure 1 is a view showing the opposite faces of a set of sixteen blocks.

Fig. 2 is a top view of a receptacle for receiving the block Fig. 3 is a transverse section through the box with the cover in place.

Figs. 4, 5 and 6 show one method of arranging the blocks in the receptacle with reference to the des' ns curried thereby.

Fig. 7 shows the b ocks of Fig. 6 turned over in the cover of the receptacle to expose the numerals.

Fig. 8 shows a somewhat different method of arrangement.

Fig. shows the blocks of F lg. 8 turned over in the cover of the receptacle.

Figs. 10 and 11 are detail views of's block with a detaclurble numeral plate.

Fig. 12 is a view showing-a method of arranging the blocks by meuns of the designs thereon to receive a new set of numorals.

Fig. 13 is a view showing the blocks of F lg. 1?. turned over into the cover of the receptocle and having a new set of numerals attached thereto.

Figs. 14- aud 15 show an arrangement of the blocks according to the designs carried thereby and the arrangement of numerals corresponding thereto.

Sixteen blocks bearing numerals l to 16 and certain characters or designs on the faces opposite the numerals as shown in Fig. 1 fit in a receptacle A which is divided into sixteen equal squares, each equul in size to one of the blocks.

Relative positions of the various block receiving squares may be determined by reference to the vertical and horizontal rows or by reference to one of these rows and a. disgoual row. Only two of the diagonal rows however include four squares. The "crticul rows, as Well as the horizontal rows, are mutually exclusive. In order to provide four mutually exclusive diagonal rows. the squares traversed by the heavy single line a are considered in one diagonal row and the squares traversed by the double lines I) are considered another diagonal row. Hereafter the expression in line with means in the some horizontal vertical or diagonal row as defined above.

As shown in Fig. l, the blocks hearing numerals l and .16 have uncolorod designs, .1 u plain triangle, and 16 the some triangle respond with two 015 the large single colored triangles. These six blocks show every 305- sible combination of two-out of four colors. The four remaining blocks 2, 3, 5 and 9 each have three colored trianglesbenring th rec of the four colors :iboi-zc named. These four blocks show all the combinations of three out of four colors.

To arrange the blocks to form a, magic sqnere, the two lend blocks Jazz and Jinx may be placed anywhere in the same diagonal line and the our single colored triangle blocks are. placed in line with Jazz but out of line with Jinx, as shown in Fig. 4. The six double triangle blocks are then laced so that each ot the two triangles is ine horizontally or vertically with the large triangle of the-some color. As shown in Fig. 5, the block having the red and yellow trian les is in the some horizontal line with the blocks having the large red and large yellow triangles, the block bearing the two triangles colored brown and green in the same vertical line with the two blocks hearing the large brown and green triangles, the block having two triangles colored brown and red is in the same horizontal linens the large brown triangle and in the same vertical line as the large red trian le, and the three remaining double triangle Blocks have similar ositions with respect to the two large triang e blocks hearing the two colors of each. There now remain four vacant places to be filled by the four blocks having three colored triangles. For each vacant place note the three. colors present in the vertical or horizontal line throu d1 the vacant space which does not inc-lure the Jinx block, and insert in each place the block bearing the three colors present in this line. In the second horizontal line of Fig. 5 the colors hmwn, red and yellow occur. The block bearing brown, red and yellow triangles fills out this line (see Fig. 6). In the first vertical line, Fig. 5, the colors red, brown and green occur and block bearing colors red, brown, and green fills out the line. In the third vertical line yellow, brown and .frrcen occur and block bearing yellow, brown and green triangles fills out the line.

Fig. 7 shows the blocks of Fig. 6 turned over into the cover B of the receptacle. It will he noted thnt each vertical. horizontal and diagonal line totals 34.

Fig. 8 shows a. somewhat more diflicult arrangement. According to this view the J azz and Jinx blocks have been placed in the some horizontal line, the four large colored trian les have been placed in line vertically an di onell with the Jazz block, the six dou 1e triangle blocks have been placed witheach triangle in line vertically or'cliagonally with the large triangle of the same color, and the four three-color blocks each fills out a. vertical or diagonal line in: which the three colors are already present.

A general statement of the rule of arrangement is that the Jazz and Jinx blocks must he elated in-the same line with each other vertically, horizontally or dingonally. The four single color blocks are then placed in line with Jane but out of line with Jinx. The six double color blocks are then placed with each of their coloredtriangles. in line with the. big triangle ofthe same color along one of the two classes. of lines other than that upon which both. Jazz and Jinx. are located; i. e. if Jazz end.J inx are on a, diagonal line the double triangle blocks are lined up. along vertical and; horizontal lines; if Jazz and J inx are on a. horizontal line the double triangle blocks are lined up along vertical and diagonal. lines, and if Jazz and Uinx are on a. vertical; line the double triangle blocks ere lined up alon horizontal: and diagonal lines. When all t e double colored blocks are lured, there will remain four: places to be -ed-with. the three-color blocks. One of the three lines through. each vacant place will contain one large triangle and two double triangle blocks, and three colors. will appear on these three blocks. The block hearing the three colors appear-ring in such a line fills out the line.

The Jazz block can be placed in any one of the sixteen squares in the box, the Jinx block can then be placedin. any one of nine squares and the four single. color blocks can be arranged in an; orderin the four squares cletermined'bythe position of the two leader blocks. Since there 24 different ways of arranging four things in four places it follows that the abovc rnle covers l6x9x24 or 3,456 different magic squnre arrangements.

It will be noted thut the location of all the blocks in the square is positively determined when the two leader blocks and the four single blocks have been placed. In Figs. 7 and 9 the blocks hearing numernls 8 and 9 have been narked A and B respectively, and the four blocks hearing numerals 1, 10, 11 and 13 have each been marked (1.. The blocks marked. A and B are lead blocks correspond ing to Jazz and Jinx and the four blocks marked (2' correspond to the four single triangle blocks. ll henever end ll are placed in line with each other and the four (1. blocks are placed in line with each other and the four a blocks are placed in line with A but out of line with B it is possible to place the remaining blocks to form a magic square. Each arrangement is a prob-- lem in itself but intelligent observation of the arrangements produced by the methods first described will enable one to quickly fill in the ten remaining blocks in their proper places for any such arrangement of the six marked blocks.

In order to make fuller use of the possibilities inherent in the system of markings above described the blocks a may be provided with detachable number plates as shown in Figs. 10 and 11. As many number plates as desired may be used.

Fig. 12 shows the blocks arranged to receive a set of numerals. In this arrangement which might be termed the reverse arrangement, the Jazz and Jinx blocks are placed in the same diagonal line with the further limitation that they must both be in squares bordering opposite sides of the box or both be in inside squares. The tour single color blocks are then placed in line with Jinx but out of line with Jazz. The double color blocks are then each lined up in vertical or horizontal lines with the two blocks having large triangles of the two colors not on the two-color block. Each of the three color blocks then fills out a line in which the one color absent therefrom occurs twice; i. e. in the second horizontal line of Fig. 10 the color green occurs twice. The block having colors brown, red, and yellow fills out this line. Under the above rule of arrangement one of the leader blocks may be placed in any one of the sixteen squares in the box. The position of the second leader block is fixed by the first, but the four single color blocks can be arranged in any one of 24 different ways. The total number of arrangements under the above rule is therefore 16 x 2% or 384. If. the numerals to 16 be applied in consecutive order to the blocks arranged according to any one of the 384 reverse arrangements, the rule of arrangement described in connection with Figs. 4 to 9 will still hold good for forming magic squares.

Other groups of numerals other than the numerals 1 to 16 may be attached to the blocks. Sixteen consecutive terms of any series which fulfills the following conditions may be selected:

The differences between successive terms are V X Y Z where W X Y and Z are each positive or ne ative integers and the series progresses as follows: W, X, W, Y, W, X, W, Z, X, W, Y, W, X, W.

Select any number for the first term, assign arbitrary values to W X Y and Z, add. W to first term to get second, X to second to get third, W to third to get fourth, and Y to fourth to get fifth, etc, as indi cated above. In Fig. 13, W equals 3, X equals 2, Y equals 2, Z equals l2. .\nothrr way of stating the rule is that when the Hi numbers are arranged in a square in rousecutive order as in Fig. 1'3 there must be a constant difference between the adjacent numerals in any two parallel columns taken vertically or horizontally, and the difference between numerals in the first and second columns must equal the difi'erence between numerals in the third and fourth columns, taking columns both vertically and horizontally. may be all equal in which case the numbers are successive terms of an arithmetic series, or any two or three of the four differences ma be equal.

"he numerals of the series shown in Fig.

13 are shown applied to the blocks in consecutive order beginning at the upper left hand corner. In Fig. 14 the blocks with the numerals attached as indicated in Figs. 12 and 13 are arranged in magic square order similar to that shown in Figs. 4 and (i. Fig. 15 shows the blocks of Fig. 14 turned over into the cover of the receptacle. The numerals as arranged in Fig. 15 total 100 on every vertical horizontal and diagonal line. After a player has learned to readily build any of the various magic square arrangements with the aid of the six marked numeral blocks only, the next problem is to ormulate a rule whereby any numeral can be placed in any square, its complemental numeral in any Of the nine squares alined with the first numeral, and the four numerals which are to be placed in line with the first numeral and out of line with its complemental numeral determined, so that any of the 3456 arrangements can be made without the aid of any system of marking. The solution of the series of problems above suggested will furnish the data necessary for a complete mathematical explanation of the puzzle.

What I claim is:

l. A. set of puzzle blocks comprising sixteen blocks bearing numerals representing successive terms of a rec-urrin series, two of said blocks having dist-inguisiliing designs thereon, and four other blocks having designs distinguishing them from the first two blocks, said four blocks bearing numerals which have a definite mathematical series relationship with the numerals on the two first mentioned blocks substantially as described.

2. A set of puzzle blocks comprising sixteen blocks, each having a distinguishing design thereon, two of said blocks bearing designs that readily distinguish them from each other and from all the other blocks,

The diiierenccs W X Y and Z four of said blocks bearing: designs which differ from each other but identify them as a group, six of, said blocks bearing designs involving different i-ombinations of distinguishing features of two of said group oi four blocks and the four remaining blocks bearing different combinations oi distin guishing features of three of said group of four blocks.

3. The set of puzzle blocks as set forth in claim 2 together with detachable and interchangeable numeral planes thenefor.

4. In (h set of puzzle bloeks, sixteen blocks bearing numerals representing; successive terms of ii recurring series two of? said blocks whose numerals have it eo nplementul reiutionship having designs thereon which distinguish them firom, each other and from all the other blocks, a grou of four blocks whose numeBa-ls have a de nite, senies relationship with respect to the two first men.- tioned blocks and which have designs there on distinguishing them from each other, but which identify them as a group, a Second set of four blocks whose numerals have av complemental relationship to the numerals on the first group of iiour blocks and which have designs thereon distinguishing themfrom each other but identifying them as u groi'lp, and a group of six blocks liaiving distinguishing designs thereon which indicate a relationship with respect to a plurality of the blocks of said two groups of four blocks.

5. A set of puzzle blocks comprising blocks hearing on one face numerals representing successive terms of 3-, recurning series, said blocks having on their faces opposite the numerals distinguishing designs having group charanteristics identifyin a plurality of groups. the designs of the individuiil blocks of: one groupihm ing characteristics indicating a mathematicali relationship with. respect to blocks ofi other of saidgroups.

In testimony whereof, I heneuntoafiix my signature.

FRANK Si GREENE. 

